Definition: Let F be a subset of a metric space X. F is called closed if whenever is a sequence in F which converges to a E X, then a E F. (i.e. F contains all limits of sequences in F) The closure of F is the set of all limits of sequences in F. Claim 1: F is contained in the clousre of F. Claim 2: The closure of F is closed. How can we prove these formally? For claim 1, I think we have to show x E F => x E cl(F). For claim 2, we need to prove that cl(F) contains all limits of sequences in cl(F). But how can we prove these? I know these are supposed to be basic facts, but my book never gives examples of how to prove these from the definitions given above...and I have no clue... Any help is appreciated!